© 2016. This version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/This paper presents a comparison between the vibration energy flow radiated by a double-deck tunnel and the one radiated by a simple tunnel when both are excited by constant or by harmonic moving loads. For both cases, the radiated energy is computed using a three-dimensional semi-analytical model of the system. The total energy radiated upwards is presented for a wide range of load speeds, when a constant moving load is considered, and for a wide range of excitation frequencies, when the excitation is a harmonic moving load. Significant differences have been obtained, first, for constant loads moving at very high speeds and, second, for harmonic loads moving at typical speeds for underground trains.Peer ReviewedPostprint (author's final draft

Arcos Villamarín, Robert

Clot Razquin, Arnau

Romeu Garbí, Jordi

Soil Dynamics and Earthquake Engineering

English

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   UPCommons Portal del coneixement obert de la UPC http://upcommons.upc.edu/e-prints   © 2016. Aquesta versió està disponible sota la llicència CC-BY-NC-ND 4.0 http://creativecommons.org/licenses/by-nc-nd/4.0/   © 2016. This version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/   An energy flow study of a double-deck tunnel underquasi-static and harmonic excitationsArnau Clota,∗, Jordi Romeua, Robert ArcosaaAcoustical and Mechanical Engineering Laboratory (LEAM). Universitat Polite`cnica deCatalunya, SpainAbstractThis paper presents a comparison between the vibration energy flow radiatedby a double-deck tunnel and the one radiated by a simple tunnel when bothare excited by constant or by harmonic moving loads. For both cases, theradiated energy is computed using a three-dimensional semi-analytical model ofthe system. The total energy radiated upwards is presented for a wide rangeof load speeds, when a constant moving load is considered, and for a widerange of excitation frequencies, when the excitation is a harmonic moving load.Significant differences have been obtained, first, for constant loads moving atvery high speeds and, second, for harmonic loads moving at typical speeds forunderground trains.Keywords: Underground vibrations, Double-deck tunnel, Moving loads,Energy flow.1. IntroductionThe rapid increase in the number of underground traffic infrastructures in heav-ily populated areas has motivated the design of new types of tunnels, such∗Corresponding authorEmail address: arnau.clot@upc.edu (Arnau Clot)Preprint submitted to Soil Dynamics and Earthquake Engineering July 9, 2016as double-deck tunnels. However, despite being constructed in many impor-tant cities worldwide, the vibration impact of subway traffic circulating along adouble-deck tunnel has not yet been properly studied.One of the most well-established models for predicting railway-induced vibra-tions in tunnels is the Pipe-in-Pipe (PiP) model presented by Forrest and Hunt[1, 2], a three-dimensional (3D) semi-analytical track-tunnel-soil model thatrepresents the tunnel-soil system as as an infinite thin cylindrical shell perfectlycoupled to a viscoelastic full-space. The model was later extended by Husseinand Hunt [3], who added a new floating-slab track model to the tunnel-soil sys-tem. The same authors also used the model to develop a power flow methodto evaluate the response of underground railway structures excited by infinitemulti-point moving loads [4]. More recently, Clot et al. [5] performed a compar-ison between the power flows radiated by a double-deck tunnel and by a simpletunnel in plane-strain conditions, finding significant differences between both.The results obtained in their work, however, do not consider the propagation ofwaves in the tunnel axial direction, ignoring how the dynamics of the interiorfloor may influence this propagation.The aim of this paper is to extend the results presented in [5] by studying theeffect that the load speed has in the comparison between the responses of double-deck and simple tunnels. In order to perform this study, the soil response tostatic and harmonic moving loads is computed using the 3D double-deck tunnelmodel recently presented in [6]. The comparison is performed by consideringthe energy of vibration radiated upwards by both tunnels.2. Analytical formulationThe proposed double-deck tunnel model is presented in Fig. 1. The tunnelstructure is modelled as an infinite thin cylindrical shell of constant thicknessht and constant mean radius rt divided into two equal parts by an interior floor,2Interior floorSoilTunnelTunnelTunnelTunnelInterior FloorrtSoilvtFFFd rxFig. 1: Representation of the double-deck circular tunnel model.which is represented as a thin strip plate of constant thickness hp and constantwidth Lp. It is assumed that the edges of the interior floor are simply supportedon the tunnel walls and that the surrounding soil is a linear homogeneous vis-coelastic full-space. Two equal vertical point loads separated a distance dr areapplied on the interior floor. The loads, which are assumed to be situated atthe same distance from the centre of the interior floor, are moving along thetunnel at a constant speed vt. The geometry and the mechanical parameters ofthe system are assumed to be invariant in the tunnel axial direction, which isconsidered as the x-direction.The soil response to a load applied on the interior floor is obtained by couplingthe interior floor subsystem with the tunnel-soil subsystem in the wavenumber-frequency domain. An upper case letter with an upper bar has been used toidentify that a variable is expressed in this domain. The coupling procedureis only outlined in the following paragraphs but the interested reader can findmore details in [6] and [7].3For coupling the interior floor to the tunnel-soil system it is assumed that, atthe tunnel-floor joint positions, the vertical displacement of the floor is equal tothe tangential displacement of the tunnel interior surface. These displacementsare expressed in terms of the strip plate transfer functions and of the PiP trans-fer functions, respectively. The PiP transfer functions are obtained using theformulation presented in [3], which extends the original PiP formulation [1] tothe case where the applied loads are antisymmetric. The analytical expressionsof the used transfer functions can be found in [6].With the considered floor-tunnel coupling conditions, the coupling forces can beexpressed in terms of the external forces F¯ , of the strip plate transfer functionsand of the PiP transfer functions. Once the coupling forces are determined, thesoil displacement field U¯ and the soil stress field T¯ are finally obtained by usingadditional PiP transfer functions. The resulting expressions can be compactlywritten asU¯i = Hu,iF¯ , T¯i = Hτ,iF¯ , (1)where Hu,i and Hτ,i are, respectively, the double-deck tunnel transfer functionsof the displacement field and of the stress field at a position i of the soil due tothe two point loads F¯ applied on the interior floor.For the case of a simple tunnel, the point loads F¯ are applied at the interiorsurface of the tunnel. As in the double-deck tunnel case, the loads are separateda distance dr and are situated at the same distance from the tunnel invert. Inthis case, the soil displacement and stress fields can be directly expressed usingEq. (1) by just replacing the double-deck tunnel transfer functions with thesimple tunnel ones.In order to compare the vibration impact that both tunnels have on nearbybuilding foundations, the energy of vibration radiated upwards by them is cal-4θ∆θθθFig. 2: Chosen integration surface for the energy flow calculation.culated [4]. The considered surface of integration is the cylindrical strip of lengthrm(θ2 − θ1) and of width ∆x represented in Fig. 2. Assuming that ∆x is verysmall and that the cylindrical strip is centered at xm = 0 m, the total energyflow E is given byE = rm ∆xθ2∫θ1∞∫−∞v(0, θ, t) · τ (0, θ, t)dtdθ. (2)where v and τ are the velocity of vibration and stress fields, respectively, causedby two unitary harmonic moving point loads.The soil velocity of vibration and stress fields caused by a unitary harmonicmoving load p(x, t) = cos(ω˜t)δ(x− vtt), where ω˜ is the excitation frequency, areobtained by first transforming it to the wavenumber-frequency domain, thenobtaining the transformed responses using Eq. (1) and, finally, transformingthese responses to the space-time domain. The resulting expressions at x = 0m arev(0, t) =1(2pi)2vtRe[∫∞−∞iωHu(ω − ω˜vt, ω)eiωtdω](3)and5τ (0, t) =1(2pi)2vtRe[∫∞−∞Hτ(ω − ω˜vt, ω)eiωtdω], (4)whereHu andHτ are the transfer functions defined in Eq. (1). Once both fieldsare known, the energy flow radiated across the considered surface is obtainedcomputing Eq. (2).3. Results and discussionThis section presents the results obtained in the comparison of the energy flowsradiated by a simple and by a double-deck tunnel for quasi-static and for a dy-namic excitations. The considered mechanical properties for the interior floor,the tunnel and the soil are presented in Tables 1 and 2. Material damping isintroduced assuming complex-valued Young modulus, in the case of the tunnelparts, and complex-valued Lame´ parameters, in the case of the soil. Both ex-ternal loads have an amplitude of 0.5 N and are separated a distance dr = 1.8m. More details regarding the computation of both tunnel models can be foundin [6].3.1. Quasi-static excitationThe external excitation considered in this section is two constant moving pointloads. The velocity of vibration and stress fields caused by constant movingloads can be obtained using ω˜ = 0 in Eqs. (3) and (4).The results presented in Fig. 3 have been obtained computing Eq. (2) at rm =10 m, for vt between 10 and 250 m/s and considering a typical Tertiary soil (a)and a soft Quaternary soil (b). The energy radiated upwards has been takeninto account defining θ1 = 0 and θ2 = pi rad, with an angular resolution of∆θ = pi/60 rad and with a space resolution ∆x = 1 m.650 100 150 200 250−160−150−140−130−120−110−100vt [m/s]Energyflow[dBref1J]50 100 150 200 250−160−150−140−130−120−110−100vt [m/s]Energyflow[dBref1J](a)(b)Fig. 3: Total energy radiated upwards by a simple tunnel (solid line) and by a double-decktunnel (dashed line) for different speeds vt. Results are presented for (a) a typical Tertiarysoil and (b) a soft Quaternary soil.7Floor Parameter Value Tunnel Parameter ValueLength (Lp) 10.9 m Radius (rt) 5.65 mWidth (hp) 0.4 m Width (ht) 0.4 mYoung modulus 27.6 GPa Young modulus 27.6 GPaPoisson ratio 0.175 Poisson ratio 0.175Density 3000 kg m−3 Density 3000 kg m−3Damping ratio 0.02 Damping ratio 0.02Table 1: Mechanical parameters used to model the interior floor as a thin plate and the tunnelas a thin shell.Soil Parameter Tertiary soil Quaternary soilYoung modulus 100 MPa 30 MPaPoisson ratio 0.3 0.3Density 1950 kg m−3 1950 kg m−3P-wave phase speed 262.74 m s−1 143.91 m s−1S-wave phase speed 140.44 m s−1 76.92 m s−1Volumetric damping ratio 0.03 0.03Deviatoric damping ratio 0.03 0.03Table 2: Mechanical parameters used to model the soils as elastic continua.Two different phenomena can be identified in the presented results. The firstone, which can be observed for both tunnels, is that a significant increase of theradiated energy is obtained for speeds between 135 and 145 m/s, in the case ofthe Tertiary soil, and between 70 and 80 m/s, in the case of the Quaternarysoil. For both soils these speed values are around their S-wave phase speed(see Table 2). The second phenomena, which is only observed in the case ofthe double-deck tunnel, is an important increase of the radiated energy whenthe speed is around 200 m/s. This second increase occurs at the same speedfor both types of soils, which indicates that it is mainly caused by the interiorfloor dynamics. However, for nowadays trains and vehicles circulation speeds,the total energy radiated upwards by both tunnels when they are excited withquasi-static loads is very similar.83.2. Harmonic excitationThe vibration energy flow radiated upwards when both tunnels are excited byharmonic moving point loads is compared in Fig. 4. The comparison has beenperformed at rm = 10 m for excitation frequencies fe = 2piω˜ between 1 Hz and80 Hz [8] with increments of 0.5 Hz and for two speeds: vt = 15 m/s (a) and 40m/s (b).The main difference between the radiated energy flows is that the double-decktunnel response presents a significant increase around 5 Hz and around 45 Hz,frequencies that are similar to those obtained in the power flow study presentedin [5]. Despite this, there are considerable differences between the ratio of energyflows presented in this work and the ratio of power flows presented there. Thesedifferences are especially clear for excitation frequencies between 50 and 80 Hz,where the results presented in Fig. 4 show that the energy flow radiated by thesimple tunnel is clearly lower than the one radiated by the double-deck tunnelwhile the previous power flow study predicted the opposite trend. Therefore,it can be concluded that a two-dimensional study can be used for estimatingthe resonance frequencies of the floor-tunnel-soil system but is not suitable forquantifying the differences between the energies radiated by both tunnels.4. ConclusionsThis paper presents a study of the vibration energy radiated by a double-decktunnel when it is excited by constant or harmonic moving point loads and com-pares this energy to the one obtained when a simple tunnel is considered.For the case of a quasi-static excitation, the total energy radiated upwards byboth tunnels has been compared for a wide range of speeds and two importantincreases in this energy have been observed. While the first increase, whichoccurs around the S-wave phase speed of the soil, has been obtained for both90 10 20 30 40 50 60 70 80−140−120−100−80−60fe [Hz]Energyflow[dBref1J]0 10 20 30 40 50 60 70 80−140−120−100−80−60fe [Hz]Energyflow[dBref1J](a)(b)Fig. 4: Total energy radiated upwards by a simple tunnel (solid line) and by a double-decktunnel (dashed line) for different excitation frequencies. Results are presented at rm = 10 mfor two loads speeds: 15 m/s (a) and 40 m/s (b).10tunnels, the other, which is not affected by the type of soil considered, hasonly been found for the case of the double-deck tunnel. However, for nowadayscirculating speeds in tunnels, the response of both systems to a quasi-staticexcitation is very similar.For the case of a dynamic excitation, the total energy radiated upwards by bothtunnels has been calculated for a wide range of excitation frequencies. Theresults show that, while smooth variations of this energy are observed in thesimple tunnel response, sharp peaks are found in the double-deck tunnel case.Therefore, significant differences have been found between the energy radiatedby both tunnels for the whole range of frequencies studied.The similarities and differences between the energy flow results shown in thiswork and the power flow ones presented in [5] have been also discussed. It isconcluded that, despite that the power flow study could estimate the resonancefrequencies of the floor-tunnel-soil system, it is necessary to take into accountthe motion of the load along the track and the 3D nature of the problem forstudying the effect of the quasi-static excitation and for quantifying the amountof energy radiated by both tunnels for the whole range of frequencies of interest.AcknowledgementsThe results presented have been obtained in the frame of the ISIBUR projectTRA2014-52718-R, ”Innovative Solutions for the Isolation of Buildings fromUnderground Railway-induced vibrations” funded by the Spanish Ministry ofEconomy and Competitiveness. This financial support is gratefully acknowl-edged.11ReferencesReferences[1] Forrest JA, Hunt HEM. A three-dimensional tunnel model for calculationof train-induced ground vibration. J Sound Vib 2006;294(4-5):678–705.[2] Forrest JA, Hunt HEM. Ground vibration generated by trains in under-ground tunnels. J Sound Vib 2006;294(4-5):706–36.[3] Hussein MFM, Hunt HEM. A numerical model for calculating vibrationfrom a railway embedded in a full-space. J Sound Vib 2007;305(3):401–31.[4] Hussein MFM, Hunt HEM. A power flow method for evaluating vibrationfrom underground railways. J Sound Vib 2006;293(3-5, SI):667–79.[5] Clot A, Romeu J, Arcos R, Mart´ın SR. A power flow analysis of adouble-deck circular tunnel embedded in a full-space. Soil Dyn EarthqEng 2014;57:1–9.[6] Clot A, Arcos R, Romeu J, Pa`mies T. Dynamic response of a double-deckcircular tunnel embedded in a full-space. Tunn Undergr Sp Tech. (to bepublished) 2016;.[7] Clot A. A dynamical model of a double-deck circular tunnel embedded ina full-space. Ph.D. thesis; Universitat Polite`cnica de Catalunya; 2014.[8] ISO 2631-2:2003 Mechanical vibration and shock. Evaluation of humanexposure to whole-body vibration. Part 2: Vibration in buildings (1 Hz to80 Hz). International Organization for Standardization; 2003.12List of Figures1 Representation of the double-deck circular tunnel model. . . . . 32 Chosen integration surface for the energy flow calculation. . . . 53 Total energy radiated upwards by a simple tunnel (solid line)and by a double-deck tunnel (dashed line) for different speeds vt.Results are presented for (a) a typical Tertiary soil and (b) a softQuaternary soil. . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 Total energy radiated upwards by a simple tunnel (solid line)and by a double-deck tunnel (dashed line) for different excitationfrequencies. Results are presented at rm = 10 m for two loadsspeeds: 15 m/s (a) and 40 m/s (b). . . . . . . . . . . . . . . . . 1013List of Tables1 Mechanical parameters used to model the interior floor as a thinplate and the tunnel as a thin shell. . . . . . . . . . . . . . . . . 82 Mechanical parameters used to model the soils as elastic continua. 814

An energy flow study of a double-deck tunnel under quasi-static and harmonic excitations

© 2016. This version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/This paper presents a comparison between the vibration energy flow radiated by a double-deck tunnel and the one radiated by a simple tunnel when both are excited by constant or by harmonic moving loads. For both cases, the radiated energy is computed using a three-dimensional semi-analytical model of the system. The total energy radiated upwards is presented for a wide range of load speeds, when a constant moving load is considered, and for a wide range of excitation frequencies, when the excitation is a harmonic moving load. Significant differences have been obtained, first, for constant loads moving at very high speeds and, second, for harmonic loads moving at typical speeds for underground trains.Peer Reviewe

An energy flow study of a double-deck tunnel under quasi-static and harmonic excitations

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